DOI: 10.2478/s11533-007-0013-5 Research article CEJM 5(3) 2007 551–580

Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues∗

Samuel Krushkal

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel and Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Received 31 July 2006 ; accepted 3 April, 2007

Abstract: The Grunsky and Teichmuller¨ norms κ(f)andk(f) of a holomorphic univalent function f in a finitely connected domain D ∞with quasiconformal extension to C are related by κ(f) ≤ k(f). In 1985, Jurgen¨ Moser conjectured that any univalent function in the disk Δ∗ = {z : |z| > 1} can be approximated locally uniformly by functions with κ(f)

Keywords: quasiconformal, univalent function, Grunsky coefficient inequalities, universal Teichmuller¨ space, subharmonic function, Strebel’s point, Kobayashi metric, generalized Gaussian curvature, holomorphic curvature, Fredholm eigenvalues. MSC (2000): 30C35, 30C62, 32G15,30F60, 32F45, 53A35

1 Introduction and main results

1.1 Grunsky inequalities and Moser’s conjecture

The classical Grunsky theorem states that a holomorphic function f(z)=z+const +O(z−1) in a neighborhood U0 of z = ∞ can be extended to a univalent holomorphic function on

∗ To Reiner Kuhnau¨ on his 70th birthday 552 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

the disk Δ∗ = {z ∈ C = C ∪{∞}: |z| > 1}

if and only if its Grunsky coefficients αmn satisfy the inequalities ∞ √ mn αmnxmxn ≤ 1, (1.1) m,n=1

where αmn are defined by ∞ f(z) − f(ζ) log = − α z−mζ−n, (z, ζ) ∈ (Δ∗)2, (1.2) z − ζ mn m,n=1 ∞ 2 2 2 2 x =(xn) runs over the unit sphere S(l )oftheHilbertspacel with x = |xn| ,and 1 the principal branch of the logarithmic function is chosen (cf. [1]). The quantity ∞ √ 2 κ(f):=sup mn αmnxmxn : x =(xn) ∈ S(l ) (1.3) m,n=1

is called the Grunsky constant (or Grunsky norm)off. Let Σ denote the collection of all univalent holomorphic functions

−1 ∗ f(z)=z + b0 + b1z + ···:Δ → C \{0}, (1.4)

and let Σ(k) be its subset of the functions withk-quasiconformal extensions to the unit {| | } 0 disk Δ = z < 1 so that f(0) = 0. Put Σ = k Σ(k). This collection closely relates to universal Teichmuller¨ space T modelled as a bounded domain in the Banach space B of holomorphic functions in Δ∗ with norm

2 2 ϕ B =sup(|z| − 1) |ϕ(z)|. (1.5) Δ∗ All ϕ ∈ B can be regarded as the Schwarzian derivatives

     2 Sf =(f /f ) − (f /f ) /2

of locally univalent holomorphic functions in Δ∗.ThepointsofT represent the functions f ∈ Σ0 whose minimal dilatation

μ μ ∗ k(f):=inf{k(w )= μ ∞ : w |∂Δ = f}

determines the Teichmuller¨ metric on T.Herewμ denotes a quasiconformal homeomor- phism of C with the Beltrami coefficient

μ(z)=∂zw/∂zw, (1.6)

and

μ ∞ = ess supC |μ(z)| < 1. (1.7) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 553

Grunsky’s theorem was strengthened for the functions with quasiconformal extensions by several authors. The basic results obtained by Kuhnau,¨ Pommerenke and Zhuravlev are as follows: for any f ∈ Σ0, we have the inequality κ(f) ≤ k(f); (1.8) on the other hand, if a function f ∈ Σ satisfies the inequality κ(f)

This was proved in [11] under the assumption that approximating maps fn are asymp- totically conformal on the unit circle S1 = ∂Δ∗.

1.2 Main theorem

Uniform convergence on compact sets is natural and sufficient in many problems of ge- ometric complex analysis. In applications of Schwarzian derivatives, especially to Teich- muller¨ spaces, one has to use the strong topology defined by the norm (1.5).

A question is: How sparse is the set of derivatives ϕ = Sf in T representing the maps with the property (1.10)? Our first main result answers this question strengthening Moser’s conjecture.

0 Theorem 1.1. The set of points ϕ = Sf , which represent the maps f ∈ Σ with κ(f)

Openness follows from continuity of both quantities κ(f)andk(f) as functions of the

Schwarzian derivatives Sf on T (cf. [12]). The main part of the proof concerns the density. The proof involves the density of Strebel points in T and relies on curvature properties of certain Finsler metrics on this space. 554 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

1.3 Application to Fredholm eigenvalues The Fredholm eigenvalues ρn of a smooth closed Jordan curve L ⊂ C are the eigenvalues of its double-layer potential, i.e., of the integral equation ρ ∂ 1 u(z)+ u(ζ) log dsζ = h(z), π ∂nζ |ζ − z| L which often appears in applications (see e.g. [5, 13–18] and the references cited there). These values are intrinsically connected with the Grunsky coefficients of the corre- sponding conformal maps. This is qualitatively expressed by the remarkable Kuhnau-¨

Schiffer theorem on reciprocity of κ(f) to the least positive Fredholm eigenvalue ρL.It is defined for any oriented closed Jordan curve L ⊂ C by

1 |D (u) −D ∗ (u)| =sup G G , ρL DG(u)+DG∗ (u) where G and G∗ are, respectively, the interior and exterior of L; D denotes the Dirichlet integral, and the supremum is taken over all functions u continuous on C and harmonic on G ∪ G∗ (cf. [10, 16]).

In general, there is only a rough estimate for ρL by Ahlfors’ inequality 1 ≤ qL, (1.11) ρL where qL is the minimal dilatation of quasiconformal reflections across L, (that is, of the orientation reversing quasiconformal homeomorphisms of C preserving L point-wise); see, e.g., [15, 19, 20]. As a consequence of Theorem 1.1,wehave

Theorem 1.2. The set of quasiconformal curves L, for which Ahlfors’ inequality (1.11) is satisfied in the strict form 1/ρL

Proof. Since all quantities in (1.11) are invariant under the action of the M¨obius group PSL(2, C)/ ± 1, it suffices to exploit quasiconformal homeomorphisms f of the sphere C 1 ∗ carrying the unit circle S = ∂Δ onto L whose Beltrami coefficients μf (z)=∂z¯f/∂zf have support in the unit disk Δ and which are normalized via (1.4), i.e., with restrictions ∗ 0 μ f|Δ ∈ Σ . Then the reflection coefficient qL equals the minimal dilatation k(w )= μ ∞ of quasiconformal extensions wμ of f|Δ∗ to C,andTheorem1.2 immediately follows from Theorem 1.1. 

In the last section we solve Kuhnau’s¨ problem related to comparing the Teichmuller¨ and Grunsky norms and establish the sharp lower bound in the inverse inequalities esti- mating k(f)byκ(f)orρL. This important problem has been open since 1981. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 555

2 Preliminaries

We briefly present here certain underlying results needed for the proof of Theorem 1.1. The exposition is adapted to our special case.

2.1 Frame maps and Strebel points

For a map f μ ∈ Σ0,wedenoteby[f μ] its equivalence class containing the maps f ν ∈ Σ0 which coincide with f μ on the unit circle S1 (and hence on Δ∗). μ0 Let f0 := f be an extremal representative of its class [f0] with dilatation

μ μ 1 1 k(f0)= μ0 ∞ =inf{k(f ):f |S = f0|S } = k, and assume that there exists in this class a quasiconformal map f1 whose Beltrami coef-

ficient μf1 satisfies the strong inequality | | ess supAr μf1 (z)

Proposition 2.1. (cf. [21]) If a class [f] has a frame map, then the extremal map f0 in this class is unique and either conformal or a Teichmuller¨ map with Beltrami coefficient μ0 = k|ψ0|/ψ0 on Δ, defined by an integrable holomorphic quadratic differential ψ on Δ and a constant k ∈ (0, 1).

This holds, in particular, if the curves f(S1) are asymptotically conformal, which includes all smooth curves.

Proposition 2.2. (cf. [22]) The set of Strebel points is open and dense in T.

The proof of this fact relies on the following lemma, which also will be used in the proof of Theorem 1.1.

Lemma 2.3. (cf. [22]) Suppose f0 with Beltrami coefficient μ0 is extremal in its class. ∞ Fix a number between 0 and 1, and take an increasing sequence {rn}1 with 0

These notions and results are extended in [22] to arbitrary Riemann surfaces; see also [36].

2.2 Basic Finsler metrics on universal Teichmuller¨ space

The universal Teichmuller¨ space T is the space of quasisymmetric homeomorphisms of the unit circle S1 = ∂Δ factorized by M¨obius maps. The canonical complex Banach structure on T is defined by factorization of the ball of Beltrami coefficients

∗ Belt(Δ)1 = {μ ∈ L∞(C): μ|Δ =0, μ < 1},

μ ν 0 letting μ, ν ∈ Belt(Δ)1 be equivalent if the corresponding maps w ,w ∈ Σ coincide on 1 ∗ S (hence, on Δ ) and passing to Schwarzian derivatives Sf μ . The defining projection

φT : μ → Swμ is a holomorphic map from L∞(Δ) to B. The equivalence class of a map wμ will be denoted by [wμ]. An intrinsic complete metric on the space T is the Teichmuller¨ metric defined by

−1 1 μ∗ ν∗ τT(φT(μ),φT(ν)) = inf log K w ◦ w : μ∗ ∈ φT(μ),ν∗ ∈ φT(ν) . (2.2) 2 It is generated by the Finsler structure on the tangent bundle T (T)=T × B of T defined by  2 −1 FT(φT(μ),φ (μ)ν)=inf ν∗(1 −|μ| ) : T ∞   (2.3) φT(μ)ν∗ = φT(μ)ν; μ ∈ Belt(Δ)1; ν, ν∗ ∈ L∞(C) . The space T as a complex Banach manifold also has invariant metrics. Two of these (the largest and the smallest metrics) are of special interest. They are called the Kobayashi and the Carath´eodory metrics, respectively, and are defined as follows. The Kobayashi metric dT on T is the largest pseudometric d on T does not get increased by holomorphic maps h :Δ→ T so that for any two points ψ1,ψ2 ∈ T,we have dT(ψ1,ψ2) ≤ inf{dΔ(0,t): h(0) = ψ1,h(t)=ψ2},

where dΔ is the hyperbolic Poincar´emetricon Δ of Gaussian curvature −4, with the differential form 2 ds = λhyp(z)|dz| := |dz|/(1 −|z| ). (2.4)

The Carath´eodory distance between ψ1 and ψ2 in T is

cT(ψ1,ψ2)=supdΔ(h(ψ1),h(ψ2)),

where the supremum is taken over all holomorphic maps h :Δ→ T. The corresponding differential (infinitesimal) forms of the Kobayashi and Carath´eodory metrics are defined for the points (ψ, v) ∈T(T), respectively, by

KT(ψ, v)=inf{1/r : r>0,h∈ Hol(Δ , T),h(0) = ψ, dh(0) = v}, r (2.5) CT(ψ, v)=sup{|df (ψ)v| : f ∈ Hol(T, Δ),f(ψ)=0}, S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 557

where Hol(X, Y ) denotes the collection of holomorphic maps of a complex manifold X

into Y and Δr is the disk {|z|

Proposition 2.4. The differential Kobayashi metric KT(ϕ, v) on the tangent bundle T (T) of the universal Teichmuller¨ space T is logarithmically plurisubharmonic in ϕ ∈ T, equals the canonical Finsler structure FT(ϕ, v) on T (T) generating the Teichmuller¨ metric of T and has constant holomorphic sectional curvature κK(ϕ, v)=−4 on T (T).

The generalized Gaussian curvature κλ of an upper semicontinuous Finsler metric ds = λ(t)|dt| in a domain Ω ⊂ C is defined by

Δ log λ(t) κ (t)=− , (2.6) λ λ(t)2

where Δ is the generalized Laplacian 2π 1 1 iθ − Δλ(t) = 4 lim inf 2 λ(t + re )dθ λ(t) r→0 r 2π 0

(provided that −∞ ≤ λ(t) < ∞). Similar to C2 functions, for which Δ coincides with the usual Laplacian, one obtains that λ is subharmonic on Ω if and only if Δλ(t) ≥ 0; hence, at the points t0 of local maximuma of λ with λ(t0) > −∞,wehaveΔλ(t0) ≤ 0. The sectional holomorphic curvature of a Finsler metric on a complex Banach manifold X is defined in a similar way as the supremum of the curvatures (2.6)over appropriate collections of holomorphic maps from the disk into X for a given tangent direction in the image. The holomorphic curvature of the Kobayashi metric K(x, v) of any complete hyper- ≥− T bolic manifold X satisfies κKX 4 at all points (x, v) of the tangent bundle (X)of

X, and for the Carath´eodory metric CX we have κC(x, v) ≤−4. For details and general properties of invariant metrics, we refer to [23, 24](seealso[28, 34]).

2.3 Grunsky coefficients revised

An underlying fact in the applications of the Grunsky coefficients to Teichmuller¨ space

theory is that these coefficients regarded as the functions of Schwarzian derivatives Sf , which we will denote by

αmn(Sf )=αmn(f), (2.7) are, together with the Taylor coefficients of f, holomorphic on T. 558 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

0 To characterize the functions f ∈ Σ obeying the property (1.10), denote by A1(Δ) the subspace of L1(Δ) formed by holomorphic functions in Δ, and consider the set

2 2 A1 = {ψ ∈ A1(Δ) : ψ = ω ,ωholomorphic} which consists of the integrable holomorphic functions on Δ having only zeros of even order. Put

μ, ψΔ = μ(z)ψ(z)dxdy, μ ∈ L∞(Δ),ψ∈ L1(Δ) (z = x + iy). D

Proposition 2.5. (cf. [7], and [20]) The equality (1.10) holds if and only if the function f is the restriction to Δ∗ of a quasiconformal self-map wμ0 of C with Beltrami coefficient μ0 satisfying the condition sup |μ0,ϕΔ| = μ0 ∞, (2.8) ∈ 2 where the supremum is taken over holomorphic functions ϕ A1(Δ) with ϕ A1(Δ) =1. If, in addition, the class [f] contains a frame map (is a Strebel point), then μ0 is of the form 2 μ0(z)= μ0 ∞|ψ0(z)|/ψ0(z) with ψ0 ∈ A1 in Δ. (2.9)

Geometrically the condition (2.8) means that the Carath´eodory metric on the holo- morphic extremal disk {φT(tμ0/ μ0 ):t ∈ Δ} in T coincides with the Teichmuller¨ metric of this space. For analytic curves f(S1) the equality (2.9) was obtained by a different method in [10].

2.4 Generalized Grunsky coefficients

The proof of Theorem 1.1 involves generic holomorphic disks in T and a new Finsler structure on T determined by generalized Grunsky coefficients. The method of Grunsky inequalities is extended to bordered Riemann surfaces X with a finite number of boundary components, in particular, to multiply connected domains on the complex plane (cf. [1, 29–31]). However, unlike the case of functions univalent in the disk, a quasiconformal variant of this theory has not been developed so far. In the general case, the generating function (1.2) must be replaced by a bilinear differential ∞ f(z) − f(ζ) − log − R (z, ζ)= β ϕ (z)ϕ (ζ): X × X → C, (2.10) z − ζ X mn m n m,n=1 where the surface kernel RX (z, ζ) relates to the conformal map jθ(z, ζ)ofX onto the sphere C slit along arcs of logarithmic spirals inclined at the angle θ ∈ [0,π)toaray issuing from the origin so that jθ(ζ,ζ)=0and − − → −1 ∞ jθ(z)=z zθ +const+O(1/(z zθ)) as z zθ = jθ ( ) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 559

∞ (in fact, only the maps j0 and jπ/2 are applied). Here {ϕn}1 is a canonical system of holomorphic functions on X such that (in a local parameter)

a a +1 ϕ (z)= n,n + n ,n + ... with a > 0,n=1, 2,..., n zn zn+1 n,n

 and the derivatives (linear holomorphic differentials) ϕn form a complete orthonormal system in H2(X). We shall deal only with simply connected domains X ∞with quasiconformal boundaries. For any such domain, the kernel RX vanishes identically on X × X, and the expansion (2.10) assumes the form

∞ f(z) − f(ζ) α − log = mn , (2.11) z − ζ f(z)mf(ζ)n m,n=1

∗ ∞ ∞ where f denotes a conformal√ map of X onto the disk Δ so that f( )= ,  f (∞) > 0, and αmn = βmn/ mn are the normalized generalized Grunsky coefficients.

These coefficients also depend holomorphically on Schwarzian derivatives Sf . A theorem of Milin extending the Grunsky univalence criterion to multiply connected domains X states that a holomorphic function f(z)=z +const+O(z−1)inaneighbor- hood of the infinite point z = ∞ can be continued to a univalent function in the whole

domain X if and only if the coefficients βmn in (2.10) satisfy the inequality ∞ 2 βmn xmxn ≤ x (2.12) m,n=1

2 for any point x =(xn) ∈ S(l )(see[30]). Accordingly, we have the generalized Grunsky constant ∞ 2 κX (f)=sup βmn xmxn : x =(xn) ∈ S(l ) (2.13) m,n=1 which coincides with (1.3)forX =Δ∗.

3 Proof of Theorem 1.1

Denote

Ge := {ϕ = Sf ∈ T : f satisfies (1.10)}, Gi := {ϕ = Sf ∈ T with κ(f)

In view of continuity of both functions κ(Sf ):=κ(f), k(Sf ):=k(f)

∗ on T, we need to establish only that each point ϕ = Sf ∗ ∈ Ge is the limit point of a

sequence {ϕn}⊂Gi. 560 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

We may assume that ϕ∗ is a Strebel point. Then its class [f ∗] contains a Teichmuller¨ extremal map with Beltrami coefficient k∗|ψ∗(z)|/ψ∗(z), |z| < 1, μ∗(z)= 0, |z| > 1,

determined by a holomorphic quadratic differential ψ∗ which has in the unit disk Δ only sμ∗ zeros of even order. By proposition 2.5, the maps f with |s| < 1/ μ ∞ also satisfy the equality (1.10), thus we can assume that

∗ ϕ ∞ < 2. (3.1)

We fix r ∈ (0, 1) and define a family of Beltrami coefficients μt = μ(·,t) depending on a complex parameter t, letting ⎧ ⎨ ∗ | | μ (z), z

and μ(z, t)=0for|z| > 1. The admissible values of t are those for which |μ(z, t)| < 1. This inequality holds, provided t ranges over the disk ∗ {  ∈ C |  | } ∗ − ∗ 2 − Δa = t : t + a >R(a) with a = a(k )=1/[1 (k ) ] > 1,R(a)=a(a 1). (3.2) Note that this disk contains the half-plane

H0 := {t ≥−1/2},

for which we have the inequality |μ(z, t)|≤k∗.Fort = ∞ and t = 0, we have, respectively, ∗ μ∞ = μ and μ∗(z)if|z| M consisting of points with the equality (1.10). The first case is trivial, thus we consider a more general situation, when

the disk (3.2) contains a sequence {tn} going to infinity and such that

μ(·,tn) μ(·,tn) lim μ(·,tn) − μ(·, ∞) ∞ =0; κ(f )= k(f ),n=1, 2,... . (3.3) n→∞ We claim that the assumption (3.3) yields that the equality

κ(f μt )= k(f μt ) (3.4) ∈ ∗ holds for all t Δa. Every class [f μ(·,t)] contains a unique extremal Teichmuller¨ map, thus the images of extremal Beltrami coefficients ∈ ∗ ϕt = φT(μt),tΔa, (3.5) S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 561

run over a holomorphically embedded disk in T;denotethisdiskbyΩa.

To establish (3.4), we construct on Ωa a Finsler metric with generalized Gaussian curvature of at most −4 and compare it with the Kobayashi metric. The underlying fact 2 is that Grunsky coefficients αmn(Sf ) generate for each x =(xn) ∈ S(l ) a holomorphic map ∞ √ hx(ϕ)= mn αmn(ϕ)xmxn : T → Δ. (3.6) m,n=1 T × ∗ Consider in the tangent bundle (T)=T B the holomorphic disks Δa(v)covering ∗  ∈ the disk φT(Δa)inT. Their points are pairs (ϕ, v), where v = φT[ϕ]μ B is a tangent vector to T at the point ϕ,andμ runs over the ball

{ ∈ C | ∗ } Belt(Dϕ)1 = μ L∞( ): μ Dϕ =0, μ ∞ < 1 . ∗ ∗ ∈ 0 Here Dϕ and Dϕ denote the images of Δ and Δ under f = fϕ Σ with Sf = ϕ. To get the maps Δ → T preserving the origin, we transform the functions (3.6)by the chain rule for Beltrami coefficients

wν = wσ(ν) ◦ (f ν0 )−1

with ν0 ν − ν0 ∂ f σ(ν) ◦ f ν0 = z , − ν0 1 ν0ν ∂zf preceded by the M¨obius map t − a(k) t → τ = (3.7) R(k) and by an appropriate self-map of Δ chosen so that ϕ∗ is obtained as the image of the origin of Δ.

Denote the composed maps by gx[σϕ] and apply them to pulling back the hyperbolic metric (2.4) onto the disks in T (T) covering the disk

{ ∈ ∗}⊂ Dσ := φT(μt): t Δa T. | | Then we obtain on covering disks the conformal subharmonic metrics ds = λgx[σϕ](t) dt with |  | ∗ gx[σϕ] λgx[σϕ] = gx[σϕ] (λhyp)= 2 , (3.8) 1 −|gx[σϕ]| having Gaussian curvature −4 at noncritical points. Consider the upper envelope of these metrics λκ(t)=supλgx[σϕ](t),

2 taking the supremum over all x ∈ S(l )andallσϕ ∈ Belt(Δ)1, and take its upper semicontinuous regularization

 λκ(t) = lim sup λκ(t ). (3.9) t→t 562 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

The last metric actually depends only on the points ϕ ∈ T and therefore it descends from ∗ ∗ the covering disks Δa(v), to an upper semicontinuous metric on the underlying disk Δa. Denote the last metric by λκ. Applying in a straightforward way the arguments exploited in [32], one obtains:

Lemma 3.1. (a) The metric λκ is a logarithmically subharmonic Finsler metric on Ωa; ∈ ∗ ∗ (b) In terms of the parameter τ related to t Δa by (3.7), with τ(t )=0, we have the equality ∗ λκ(τ)=κ(τϕ )+o(τ) as τ → 0, (3.10) ∗ 0 ∗ where κ(τϕ ) denotes the Grunsky constant of the map f ∈ Σ with Sf|Δ∗ = τϕ .

Another important property of this metric is given by the following lemma.

≤− Lemma 3.2. The generalized Gaussian curvature of λκ satisfies kλκ 4.

The last inequality is equivalent to the following one

2 Δ log λκ ≥ 4λκ,

2uκ or Δuκ ≥ 4e ,whereuκ =logλκ.HereΔ again means the generalized Laplacian. Proof of Lemma 3.2 (cf. [23, 32]). Take a maximizing sequence of (renormalized) functions (3.6) for κ(ϕ∗), so that

∗ ∗ lim hx(p) (ϕ )=κ(ϕ ), p→∞

(p) and construct the corresponding maps gx [σϕ]. Restrict these maps to the disk Ωa and apply again the parameter τ ranging over Δ. The above restrictions converge uniformly  ∗ on compact subsets in Δ to a holomorphic map g0(τ): Δ→ Δ such that g0(0) = κ(ϕ ).

This map determines on Ωa the conformal metric

 |g0(τ)| λg0 (τ)= 2 1 −|g0(τ)| of constant curvature −4 at its noncritical points; in view of (3.10) and the definition of ≤ λκ,itisasupporting metric for λκ at τ = 0 (i.e., λg0 (0) = λκ(0) and λh0 (τ) λκ(τ) near τ =0). Hence, log λh0 has a local maximum at 0, and therefore, λκ λ h0 − ≤ Δ log (0) = Δ log λh0 (0) Δ log λκ(0) 0, λκ which yields Δ log λκ(0) Δ log λ (0) − ≤− h0 2 2 , (3.11) λκ(0) λh0 (0) ≤−  and the desired inequality κλκ 4onΩa follows. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 563

Let us compare now λκ with the infinitesimal Kobayashi metric λK of T restricted to

Ωa which is logarithmically subharmonic and has generalized Gaussian curvature −4on this disk. For a fixed ϕ0 ∈ T,weset

hx(ϕ) − hx(ϕ0) Hx(ϕ; ϕ0)= 1 − hx(ϕ0)hx(ϕ)

and define similarly to (2.3) the Finsler structure Fκ(ϕ0,v)onT (T)by

2 Fκ(ϕ0,v)=sup{|dHx(ϕ0; ϕ0)v| : x ∈ S(l )}. (3.12)

It is dominated by the canonical Finsler structure (2.3). The structure (3.12) allows us to construct in a standard way on embedded holomorphic disks γ(Δ) the Finsler metrics λ(t)ds by  λγ(t)=Fκ(γ(t),γ(t)). and, accordingly, the corresponding distances

dγ(ϕ1,ϕ2)=inf λγ(t)dst, (3.13)

β

1 taking the infimum over C smooth curves β :[0, 1] → T joining the points ϕ1 and ϕ2.

Lemma 3.3. On any extremal Teichmuller¨ disk Δ(μ0)={φT(tμ0): t ∈ Δ} (and its isometric images in T), we have the equality

r −1 tanh [κ(Sf rμ0 )] = λκ(t)dt. (3.14) 0

μ0 Proof. Put f0 = f and consider the covering maps hx(μ)=hx ◦ φT : Belt(Δ)1 → Δ of (3.6)forϕ ∈ Δ(μ0). For any appropriate hx, we have the equalities

hx()  −1 |dt| tanh [hx()] = = λ (t)|dt|, (3.15) 1 −|t|2 hx 0 0 and therefore,

  −1 κ 0 | | | | tanh [ (f )] = sup λhx (t) dt = sup λhx (t) dt . (3.16) x x 0 0

2 The second equality in (3.16) is obtained by taking a sequence {xn}⊂S(l )andthe corresponding monotone increasing sequence of metrics

λ1 = λ ,λ2 =max(λ ,λ ),λ3 =max(λ ,λ ,λ ), ... hx1 hx1 hx2 hx1 hx2 hx3 564 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 so that lim λn(t)=supλhx (t). n→∞ x

Using the continuity of κ(Sf )onT and the properties of quasiconformal maps, one can show that λκ = λf0 is upper semicontinuous on Δ (cf. [39]). Thus the equality (3.14) follows from (3.9)and(3.16), and the lemma follows. 

The last step in the proof of Lemma 3.3 can be simplified. Indeed, since the upper semicontinuous regularization of sup λhx can decrease the function, we get from (3.15) x

t | |≤ −1 κ λκ(f0)(t) dt tanh [ (f0)]. 0 ≤ But for every hx,wehaveλhx (t) λκ(f0)(t), which yields the opposite inequality.

∗ Now, taking into account that the disk Ωa touches at the points ϕ and ϕn = ∗ φT(μ(·,tn)) the Teichmuller¨ disks Δ(μ )andΔ(μn) and that the metric λκ does not depend on the tangent unit vectors whose initial points are the points of Ωa,oneobtains from Lemma 3.3 and (1.8) that this metric relates to the Kobayashi metric λK|Ωa as follows ∗ λκ(0) = λκ∗ (ϕ )=λK(0),λκ(t )=λκ (ϕ )=λK(t ); n n n n (3.17) λκ(t) ≤ λK(t) for all t ∈ Ωa \{0,tn} which means that λκ is a supporting metric for λK|Ωa at t =0andt = tn,n=1, 2,... . A more subtle comparison of these metrics is obtained by applying Minda’s maximum principle:

Lemma 3.4. (cf. [33]) If a function u :Ω→ [−∞, +∞) is upper semicontinuous in a domain Ω ⊂ C and its generalized Laplacian satisfies the inequality Δu(z) ≥ Ku(z) with some positive constant K at any point z ∈ Ω,whereu(z) > −∞,andif

lim sup u(z) ≤ 0 for all ζ ∈ ∂Ω, z→ζ then either u(z) < 0 for all z ∈ Ω or else u(z)=0for all z ∈ Ω.

For a sufficiently small neighborhood U0 of the origin t = 0, we put

M = {sup λK(t):t ∈ U0}; then in this neighborhood, λK(t)+λκ(t) ≤ 2M. Consider the function

λκ u =log . λK

Then (cf. [23, 33]) for t ∈ U0,

2 2 Δu(t)=Δ log λκ(t) − Δ log λK(t)=4(λκ − λK) ≥ 8M(λκ − λK). S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 565

The elementary estimate

M log(t/s) ≥ t − s for 0

(with equality only for t = s) implies that λ (t) g0 ≥ − M log λg0 (t) λd(t), λd(t) and hence, Δu(t) ≥ 4M 2u(t).

Applying Lemma 3.4, we obtain that, in view of (3.17), both metrics λκ and λK must be

equal in the entire disk Ωa, which is equivalent to the desired equality (3.4). In fact, the above arguments give more, namely, that

κ(ϕ)=k(ϕ) (3.18)

on the connected component Π0 of the intersection

Π={tϕ∗ : t ∈ C}∩T

containing the origin of T (which is simply connected by Zhuravlev’s theorem, see [4, 5]). In particular, the last equality holds also at −ϕ∗. The same arguments work for intersections of T with arbitrary complex lines passing through ϕ∗, which yields that (3.18) must hold for all points of a ball B(ϕ∗,δ) centered at ϕ∗. ∗ ∗ ∗ Moving this ball from the point ϕ along the segment [−ϕ ,ϕ ] ⊂ Π0, one derives that the equality (3.18) must hold for all points of a ball centered at the origin ϕ = 0.Butthe

latter is impossible, since it contradicts the existence of points ϕ = Sf in a neighborhood of 0 at which κ(f)

A straightforward modification of the above arguments yields the following result.

Theorem 3.5. Let Ω=h(Δ) be a holomorphically embedded disk in T. Assume that there exists a sequence {ϕn} of points Ω convergent to ϕ0 ∈ Ω so that κ(ϕn)=k(ϕn),n=0, 1,... .

Then κ(ϕ)=k(ϕ) for all ϕ ∈ Ω.

4 Grunsky’s norm of frame maps

The following theorem is a corollary of the proof of Theorem 1.1.

Theorem 4.1. If a function f ∈ Σ0 has the property κ(f)=k(f), then for an arbitrary μr fixed r ∈ (0, 1) its frame maps f with μr given by (2.1) either simultaneously have this property or none of them does this. 566 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

Proof. The proof follows in the same manner as the first stage of the proof of Theorem 1.1. Assuming that we have for some r ∈ (0, 1) the equality κ(f μr )=k(f μr ), we can consider again the same disk Ωa ∈ T and, applying the above arguments to the pair ∗ (ϕ ,ϕr = Sf μr ), obtain that κ(ϕ)=k(ϕ) for all ϕ ∈ Ωa. 

5 Inversion of Grunsky and Ahlfors inequalities

5.1 Sharp lower estimate

We now turn to the open problem on the sharp lower estimates in the inequalities inverse to (1.8)and(1.11). Such estimates are important for example in algorithms for finding the exact or approximate values of Fredholm eigenvalues and reflection coefficients of curves (cf. e.g. [6, 14–17, 19, 20, 39]). We provide a new approach which involves the conformal metrics of negative integral curvature and certain geometric features of the universal Teichmuller¨ space. As mentioned in Section 1, there is an explicit bound k1(k) for dilatations of quasi- conformal extensions of f ∈ Σwithκ(f) ≤ k found in [40]. It is given by 1+k1 3/2 1+κ 1+κ K1 := ≤ max λ , 2λ − 1 , (5.1) 1 − k1 1 − κ 1 − κ where λ(K)=maxw(1), taking the maximum among all K-quasiconformal automorphisms w of C with the fixed points −1, 0, ∞. The distortion function λ can be represented by elliptic integrals (see [41], p. 15). For small κ < 1/3, there is a somewhat better estimate

k<3κ, (5.2) (see [6, 15]). Neither of the bounds (5.1)and(5.2)issharp. The following theorem yields a sharp bound for an individual function and improves Proposition 2.5. We shall use here the following notations. For an element μ ∈ Belt(Δ)1 we define μ(z) μ∗(z)= μ ∞ ∗ so that μ ∞ = 1. The extremal Beltrami coefficient in the class [f] will be denoted by μ0(z; f). For a measurable real valued function u on the disk Δ which is locally bounded from above we define its circular mean Mu by 2π 1 Mu(r)= u(reiθ)dθ. (5.3) 2π 0 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 567

By Jensen’s inequality, for any convex function ω on interval I ⊂ R containing the values of both u and Mu,wehave ω(Mu) ≤Mω(u). We will apply this inequality for ω(u)=eu and ω(u)=tanh−1 u. The mean (5.3) inherits certain important properties of its original function. For example, if u is subharmonic on Δ, then so is Mu.

Theorem 5.1. For every function f ∈ Σ0, we have the sharp bound κ(f) 1 k(f) ≤ = , (5.4) α(f) α(f)ρ(f) where ∗ α(f)= inf sup μ0(z; f)ψ(z)dxdy > 0. (5.5) μ0 ∈[ ] 2 f f ψ∈A ,ψA =1 1 1 Δ If f has a unique extremal extension f μ0 ,then 1 k(f μ0 ) ≤ min κ(f tμ0 ). (5.6) α(f μ0 ) |t|=1 with μ0 ∗ α(f )= sup μ0(z; f)ψ(z)dxdy > 0. (5.7) 2 ψ∈A ,ψA =1 1 1 Δ

tμ0 Note that (5.6) is a simple consequence of (5.4), because k(f )= μ0 ∞ for all t ∈ S1. This is true, in particular, for Strebel points. On uniqueness of extremal maps with nonconstant dilatations see [35]. The quantity 1 − α(f) can be regarded as a measure of deviation of the Grunsky structure Fκ defined above from the canonical Finsler structure FT on T (T). For any f satisfying (1.10), α(f)=1. The applications of this theorem will be given in the next section.

5.2 Preliminaries: some other generalizations of Gaussian curvature, and circularly symmetric metrics

The proof of Theorem 5.1 involves certain known results on conformal metrics

ds = λ(z)|dz|

on the disk Δ with λ(z) ≥ 0 (called also semi-metrics) of negative integral curvature bounded from above. We already dealt with nonsmooth conformal Finsler metrics satisfying the inequality

Δ log λ ≥ Kλ2, (5.8)

where Δ meant a generalization of the Laplacian 4∂∂. 568 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

Another generalization of this condition is due to Ahlfors [13]. A conformal metric λ|dz| in a domain G on C (or on a Riemann surface) has curvature less than or equal to  K in the supporting sense if for each K >Kand each z0 with λ(z0) > 0, there is a 2 C -smooth supporting metric λ for λ at z0 (i.e., such that λ(z0)=λ(z0)andλ(z) ≤ λ(z) ≤  in a neighborhood of z0)withκλ(z0) K (cf. [37]). There are also integral generalizations of the inequality (5.8)(seee.g.[42, 43]). We shall use its generalization in the potential sense due to [43] and say that λ has curvature at most K in the potential sense at z0 if there is a disk U about z0 in which the function 2 log λ +PotU (λ ), where Pot denotes the logarithmic potential U 1 Pot h = h(ζ)log|ζ − z|dξdη (ζ = ξ + iη), U 2π U is subharmonic. One can replace U by any open subset V ⊂ U, because the function 2 2 PotU (λ ) − PotV (λ )isharmoniconU. Note that having curvature at most K in the potential sense is equivalent to λ satisfying (5.8) in the sense of distributions. The following three preliminary lemmas were proven in [43].

Lemma 5.2. If a conformal metric has curvature at most K in the supporting sense, then it has curvature at most K in the potential sense.

The following lemma concerns the circularly symmetric (or radial) metrics which are the functions of r = |z|. Its proofs involves the fact that for the disk Δ the potential PotΔ commutes with the average (5.2).

Lemma 5.3. If a circularly symmetric conformal metric λ(|z|)|dz| in the unit disk has curvature at most −4 in the potential sense, then a λ(r) ≥ , (5.9) 1 − a2r2 where a = λ(0).

The right hand-side of (5.9) defines a supporting conformal metric for λ at the origin with constant Gaussian curvature −4onthewholediskΔ.

Lemma 5.4. Let λ|dz| be a conformal metric on the unit disk which has curvature at most −4 in the potential sense. Then the metric λ = eMu,whereu =logλ, also has has curvature at most −4 in the potential sense.

5.3 Proof of Theorem 5.1

The case κ(f)=k(f) is trivial, thus we have to deal only with the maps for which κ(f)

Consider first the case when the class [f] of a given map f is a Strebel’s point and thus [f] has a unique extremal extension f μ0 . Then on the extremal disk

∗ ∗ Δ(μ0)={φT(tμ0): t ∈ Δ}, the infinitesimal Kobayashi metric λK of T is equal to hyperbolic metric (2.4). We can assume also that k(f)= μ0 ∞ is sufficiently small so that

κ(f)

tμ∗ ∈ Put ϕt = Sf 0 for any t Δ and construct the holomorphic maps ∞ √ hx(t):=hx(ϕt)= mn αmn(ϕt) xmxn :Δ→ Δ, (5.10) m,n=1

∗ 2 2 μ0 taking again x =(xn) ∈ S(l ); then sup {|hx(t)| : x ∈ S(l )} = κ(f ). ∗ These maps determine on Δ(μ0) the corresponding conformal metrics by pulling back the hyperbolic metric (2.4):

| | ∗ hx(t) λ (t):=h (λhyp)= . hx x 2 1 −|hx(t)|

2 κ { ∈ } Now take the upper envelope of these metrics λ (t)=sup λhx (t): x S(l ) and its upper semicontinuous regularization

 λκ(t) = lim sup λκ(t ). (5.11) t→t

Similarly to the metric implied in the proof of Theorem 1.1, λκ is logarithmically subhar- monic on Δ and has the generalized Gaussian curvature and curvature in the supporting sense both less than or equal −4. By Lemma 5.2, its curvature in the potential sense is also at most −4, and by Lemma 5.4 its mean M[λκ](|t|) is a circularly symmetric metric with curvature also at most −4inthepotentialsense. μ 0 To calculate the value M[λκ](0) = λκ(0), we apply a variation of f ∈ Σ . For small μ ∞,wehave 1 ∂ f μ(ζ) 1 μ(ζ) f μ(z)=z − ζ dξdη = z − ξdη + O( μ 2 ) π ζ − z π ζ − z ∞ |ζ|<1 |ζ|<1 ∞ (5.12) b z n = + n 0 z with 1 n−1 2 b = μ(ζ)ζ dξdη + O( μ ), μ ∞ → 0. (5.13) n π ∞ |ζ|<1

The bound of the remainder in (5.12) is uniform on each disk ΔR = {|z|≤R<∞}. 570 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

The relations (1.2)and(5.13) yield the equality −1 m+n−2 2 αmn(φT(μ)) = −π μ(z)z dxdy + O( μ ∞), μ ∞ → 0. (5.14) Δ

Note also that applying Parseval’s equality to holomorphic functions ∞ k 2 2 ω(z)= ckz ∈ L2(Δ) yields for their squares ψ = ω ∈ A1(Δ) the representation 0

∞ 1 √ ψ(z)= mn x x zm+n−2, (5.15) π m n m+n=2 ∈ 2 with x =(xn) l , x = ω L2 .Wetake x =1toget ψ A1 =1. ∗ We lift the maps (5.10) from the disk Δ(μ0) onto its covering disk ∗ ∗ ∗ Δ(μ0):={tμ0 : |t| < 1} in the ball Belt(Δ)1, getting the maps hx = hx ◦ φT|Δ(μ0) whose differential at zero is given by ∞ √ ∗ 1 ∗ m+n−2 dhx(0)μ = − μ (z) mn x x z dxdy. 0 π 0 m n Δ m+n=2

Comparison with (1.3), (5.14)and(5.15) yields

μ0 M[λκ](0) = λκ(0) = α(f0),f0 := f , (5.16) where α(f0) is represented by (5.7). Then by Lemma 5.3,

M ≥ α(f0) [λκ](r) 2 2 , (5.17) 1 − α(f0) r

We claim that α(f0) > 0. (5.18) Indeed, for small |t|,wehavefrom(5.12)

tμ∗ t ∗ 2 f 0 (z)=z − μ ,ω Δ + O(t ), π 0 z where

ωz(ζ)=1/(ζ − z),ζ∈ Δ,z∈ C.

∗ 2 For any z ∈ Δ , the function ωz does not vanish in Δ and thus belongs to A1. If α(f μ0 ) = 0, we get ∗ μ0,ωzΔ =0 forall z ∈ Δ,

n which yields that μ0 is orthogonal to all powers ζ ,n=0, 1,..., and therefore to all ∗ integrable holomorphic functions in Δ (in other words, the functions ωz with z ∈ Δ span the whole space A1(Δ)). This means that μ0 is a locally trivial Beltrami coefficient, which is impossible for extremal Beltrami coefficients (see e.g. [22, 38]), and (5.18) follows. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 571

Now, integrating both sides of (5.17) over a radial segment [0,]with = μ0 ∞,one obtains

 ∗ −1 μ0 −1 μ0 μ −1 μ0 μ0 M[λκ](r)dr ≥ tanh [α(f )]=tanh [α(f )k(f 0 )] = tanh [α(f )k(f )]. 0 (5.19) Applying Lemma 3.3 yields

t −1 μ0 λκ(f μ0 )(t)|dt| =tanh [κ(f ]; 0

iθ thus after averaging λκ(re ),

  2π 2π 1 iθ 1 iθ −1 μ0 M[λκ](r)dr = λκ(re )dθdr = λκ(re )drdθ =tanh [κ(f )]. 2π 2π 0 0 0 0 0 (5.20) Comparison of (5.19)and(5.20) yields the inequality (5.6) and proves the theorem for Strebel points. Now the left-hand inequality for arbitrary f ∈ Σ0 is a simple corollary of (5.6), and the second equality in (5.4) follows then from the Kuhnau-Schiffer¨ theorem. We need to be sure that the quantity (5.5) is positive. This will be shown in the next section, but it follows also from (5.1)or(5.2). Theorem 5.1 is proved completely.

5.4 Geometric picture

A geometric meaning of the quantity (5.7) is that this value equals the supremum of L∞-lengths of projections of the unit Teichmuller¨ tangent vector

 ∗ ϕ0 = φT(0)μ0

∗ to the disk Δ(μ0) at the origin onto the elements of the distinguished set 2 A1 ∩{ψ ∈ A1(Δ), ψ =1}. Therefore, α(f) does not depend on the choice of extremal ∗ μ0 in a class [f]. More precisely, one can write

{|  | ∈ 2 } α(f)=sup νϕ0 ,ψ Δ : ψ A1, ψ A1(Δ)=1 ,

with 1 2 2 4 ν (z)= (1 −|z| ) ϕ0(1/z¯)1/z¯ , (5.21) ϕ0 2

| | tμ∗ noting that, for sufficiently small t , the Schwarzian derivative ϕ0(t):=S 0 determines f ∗ by (5.21) the harmonic Beltrami coefficient of the Ahlfors-Weill extension of the map f tμ0 across the unit circle S1. 572 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

In view of the characteristic property of extremal Beltrami differentials, we have for ∗ any such μ0 the equality ∗ ∈ ⊥ νϕ0 = μ0 + σ0,σA1(Δ) , where

⊥  A1(Δ) = {ν ∈ Belt(Δ)1 : ν, ψΔ =0forallψ ∈ A1(Δ)} =kerφT(0) is the set of so-called locally trivial Beltrami coefficients (see e.g. [22, 38]).

5.5 Generalization

The arguments applied in the first step of proof of Theorem 5.1 can be extended in a straightforward manner to any hyperbolic simply connected domain D containing the infinite point and bounded by quasicircle. Denote by Σ0(D) the class of univalent holo- morphic functions in D with expansion f(z)=z +const+O(1/z)nearz = ∞ and having 2 quasiconformal extension across the boundary of D.ThesetsA1(D)andA1(D)are defined similarly to the case of the unit disk. Then we obtain

Theorem 5.5. Let D be a quasidisk containing z = ∞. Then for every function f ∈ Σ0(D) having a unique extremal quasiconformal extension to the complementary domain C \ D, there is a sharp bound κ (f) k(f) ≤ D , αD(f) 0 where κD(f) is the generalized Grunsky constant (2.13)forΣ (D),and ∗ αD(f)= sup μ0(z; f)ψ(z)dxdy. 2 ψ∈A1(D),ψA (D)=1 1 D

5.6 Question

Is the function α(Sf )=α(f) plurisubharmonic on T ? Were the answer affirmative, we would get several interesting consequences.

6 Comparing the Teichmuller¨ and Grunsky dilatations: Kuhnau’s¨ problem

6.1 Sharp lower bound

Theorem 5.1 is rich in applications. It allows us to solve an old problem of Kuhnau¨ related to comparing the Teichmuller¨ and Grunsky dilatations. A case in point is the map t 2/3 f3 (z)=z 1+ ∈ Σ(|t|), 0 ≤|t| < 1, (6.1) ,t z3 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 573

whose extremal extension to Δ is |z| 3 2/3 f3 (z)=z 1+t ,t z with Beltrami coefficient

3 | | μ (z):=μf3,t (z)=t z /z. iθ −iθ In polar coordinates, μ3(re )=te .Themap(6.1) has threefold rotational symmetry

f(e2nπi/3z)=e2nπi/3f(z),n=0, 1, 2.

3/2 2/3 Note that f3,t(z)=ft(z ) ,where z + t/z, |z|≥1, ft(z)= z + tz, |z| < 1.

In 1981, Kuhnau¨ [9], applying the technique of Fredholm eigenvalues, discovered that the map (6.1)satisfies

κ(f3,t)

Problem. Is the exact bound in (5.2) attained by the function (6.1)? The following theorem solves this problem affirmatively and improves the estimates (5.1)and(5.2).

Theorem 6.1. For f ∈ Σ0 we have the bound 3 k(f) ≤ √ κ(f)=1.07 ...κ(f) (6.2) 2 2

(and accordingly for Fredholm eigenvalues) which is asymptotically sharp as κ → 0,with equality for the map (6.1).

In view of the density of Strebel points and continuity of k(f)andκ(f)onT,in order to prove Theorem 6.1, it suffices to establish the estimate (6.2) for Strebel points (classes) [f]. We know that each such class [f] contains a unique extremal Teichmuller¨ map f μ0 with Beltrami coefficient of the form

μ0(z)=k|ϕ0(z)|/ϕ0(z),ϕ0 ∈ A1(Δ) \{0}. (6.3)

∗ We will again use the normalized Beltrami coefficients μ0(z; f)=|ϕ0(z)|/ϕ0(z) and denote ∗ μ3(z)=|z|/z. 574 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

It follows from Theorem 5.1 that the least admissible factor in the estimate

k(f) ≤ Mκ(f) for all f ∈ Σ0 is given by

1 ∗ | Δ| =inf∗ sup μ0,ψ . (6.4) M μ0 ∈ 2   =1 ψ A1, ψ A1 We precede the proof of Theorem 6.1 by two lemmas.

Lemma 6.2. For f = f3,t, we have √ ∗ ∗ 2 2 sup |μ3,ψΔ| =max |μ3,ψ3Δ| = , (6.5) ∈ 2   =1 ψ3 3 ψ A1, ψ A1

where ψ3 are the squares of nonconstant linear functions

2 2 ψ3(z)=ω1(z) := (a0 + a1z)

  with a0 =0,a1 =0,and ψ3 A1 =1.

Proof. For any

∞ 2 2 k 2 2 2 ψ(z)=ω(z) := akz = a0 +2a0a1z +(a1 +2a0a2)z + ···∈A1 0

with nonzero a0 and a1,wehave ∗ |z| 2 2 μ ,ψΔ = [a0 +2a0a1z +(a +2a0a2)z + ...]dxdy 3 z 1 Δ 1 2π (6.6) −iθ 2 iθ 4π = [a0e +2a0a1r +(a +2a0a2)re + ...]dθrdr = a0a1. 1 3 0 0 n ∞ Noting that, by Parseval’s equality for orthonormal system { (n +1)/πz }0 ,

∞ | |2 2 an ψ A1(Δ) = ω1 L2(Δ) = π =1, 0 n +1

one derives from (6.5)and(6.6) the equality

∗ sup |μ3,ψΔ| =2 sup |a0a1|. (6.7) ∈ 2   =1 | |2+| |2 2≤1 ψ A1, ψ A1 a0 a1 / /π

2 2 The function F (a0,a1)=a0a1 is holomorphic on the closed ball

2 a1 2 2 |a1| 1 Bπ = a = a0, √ ∈ C : |a0| + ≤ , 2 2 π S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 575

√ 2 thus |a0a1| attains its maximal value on the boundary sphere Sπ = {|a| =1/ π}.This yields

2 ∗ 2 16 2 2 2 1 |a1| 8 sup |μ3,ψΔ| = π max |a0a1| =max|a1| − = , ∈ 2   =1 9 Sπ |a1| π 2 9 ψ A1, ψ A1 and this maximum is attained at the points a =(a0,a1/2) with √ √ |a0| =1/ 2π, |a1| =1/ π.

∗ If in (6.6)atleastoneofa0,a1 vanishes, then μ3,ψΔ = 0, which shows that such ψ bring no contribution to the supremum in (6.5). The lemma is proved. 

The next lemma is a slight extension of the previous one to the maps

(m+2)/2 2/(m+2) fm+2,t(z)=ft(z ) ,m=3, 5, 7 ..., whose normalized Beltrami coefficients in Δ are

∗ | |m m μm+2(z)= z /z .

Lemma 6.3. For f = fm+2,t with m =3, 5,..., we have the estimate

| ∗  |≥ | ∗  | α(fm+2,t)= sup μm+2,ψ Δ max μm+2,ψm Δ ∈ 2   =1 ψm ψ A1, ψ A1 √ (6.8) (m +1)(m +3) 2 2 ≥ > , m +2 3 where ψm are polynomials 2 (m−1)/2 (m+1)/2 2 ψm(z)=ωm(z) := a(m−1)/2z + a(m+1)/2z with 2 2 2 |a(m−1)/2| |a(m+1)/2| 1 ψ (Δ) = ω = + = . m A1 m L2(Δ) m +1 m +3 2π

Proof. Now we have to estimate | |m ∗ z 2 (m−1) m 2 (m+1) μ ,ψ Δ = a z +2a( −1) 2a( +1) 2z + a z dxdy m+2 m zm (m−1)/2 m / m / (m+1)/2 Δ 4π = a( −1) 2 a( +1) 2 m +2 m / m / for (a(m−1)/2,a(m+1)/2) satisfying

2 2 |a( −1) 2| |a( +1) 2| 1 m / + m / = . m +1 m +3 2π 576 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

The same arguments as in the proof of Lemma 6.2 yield the first inequality in (6.8). The | ∗  | maximum of μm,ψm Δ is attained when 1 m +1 1 m +3 |a( −1) 2| = , |a( +1) 2| = . m / 2 π m / 2 π Thereafter, one easily verifies by induction that the function

(m +1)(m +3) g(m)= ,m=1, 3, 5,..., (m +2)2 is strictly monotone increasing with m, which yields the second inequality in (6.8).

∗ Proof of Theorem 6.1. It remains to compare the values μ0,ψΔ for an arbitrary  ∗  extremal coefficient (6.3)with μm+2,ψ Δ. Without loss of generality, we can restrict ourselves to coefficients μ0 whose defining quadratic differentials ϕ0 are of the form

m m+1 ϕ0(z)=cmz + cm+1z + ..., m≥ 1odd,cm =0 , that is, having zero of odd order at the origin, because for zero at z0 = 0 one can take

∗  2 ∗  2 γ ϕ0 =(ϕ0 ◦ γ)(γ ) ,γψ =(ψ ◦ γ)(γ ) , with γz =(z − z0)/(1 − z0z), and this transform preserves the value of ||ϕ0|/ϕ0,ψΔ|.Then cm+1 | | | |m 1+ z + ... ∗ ϕ0(z) z cm ∗ μ0(z; f0)= = c cm+1 =: cμm + μ, 0 m ϕ (z) z 1+ cm z + ... where c = |cm|/cm and the remainder μ is of the form ∞  iθ inθ μ(re )= Cn(r)e . (6.9) −∞  −imθ The notation meansthatthisseriesdoesnotcontainthetermC−m(r)e .

If the coefficient Cm(r)in(6.9) is real and distinct from zero, then we take the poly- nomial ψm with 1 m +1 i m +3 a( −1) 2 = ,a( +1) 2 = , m / 2 π m / 2c π getting

κ μ0 ≥| ∗  | | ∗    | (f ) μ0,ψm Δ = cμm+2,ψm Δ + μ, ψm Δ 1 (m +1)(m +2) (m +1)(m +2) = i + A > > κ(f3), m +2 m +2 m m +2 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 577 where 1

Am =2π Cm(r)rdr 0 is also real and distinct from zero, and f3 = f3,t.

If Cm(r)=iβ(r),β(r) ∈ R \{0},wetakeψm with coefficients 1 m +1 1 m +3 a( −1) 2 = ,a( +1) 2 = m / 2 π m / 2c π

μ0 and obtain in a similar way the inequality κ(f ) > κ(f3).

If Cm(r)=α(r)+iβ(r)withα(r) =0 ,β(r) =0,wetake ψm with 1 m +1 m +3 a( −1) 2 = ,a( +1) 2 = m / 2 π m / 2c π and appropriate ∈ S1 so that (m +1)(m +2) (m +1)(m +2) + α(r)+iβ(r) > . m +2 m +2

μ0 In this case we again obtain κ(f ) > κ(f3).

Finally, in the case Cm(r) ≡ 0wehaveforeachψm the equality

| ∗  | | ∗  | μ0,ψm Δ = μm+2,ψm Δ ,

μ0 and κ(f ) ≥ κ(f3). Together with (6.4) this yields the assertion of Theorem 6.1. 

As observed by Kuhnau,¨ we have for the map (6.1), letting k = k(f), κ = κ(f), by [15], formula (21) (where r2 = k), k2 3κ k 1+ ≤ √ =: κ∗, 9 2 2 or equivalently, 9 9 3(κ∗)2 k2 ≤ 3 (κ∗)2 + − = ; 4 2 κ∗ 2 9 3 ( ) + 4 + 2 hence, ∗ κ ∗ 1 ∗ 2 k ≤ = κ 1 − (κ ) + ... , (κ∗)2 1 1 18 9 + 4 + 2 which shows that the equality in (6.2) is attained by the map (6.1) only asymptotically as κ → 0. 578 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

6.2 Geometric applications

Theorem 6.1 has interesting geometric consequences. The inequalities (1.8)and(6.2)resultin 3 κ(f) ≤ k(f) ≤ √ κ(f), (6.10) 2 2 and similarly for reciprocals of Fredholm eigenvalues of quasicircles f(S1).

Since κ(f) ≤ cT(0,Sf ) and the universal Teichmuller¨ space T is a homogeneous domain, one obtains from (6.10) the following inequalities estimating the behavior of invariant metrics on this space.

Theorem 6.4. For any two points ϕ1,ϕ2 ∈ T,theirCarath´eodory and Kobayashi dis- tances are related by −1 tanh dT(ϕ1,ϕ2) √3 1 ≤ −1 ≤ , tanh cT(ϕ1,ϕ2) 2 2 or 3 −1 cT(ϕ1,ϕ2) ≤ dT(ϕ1,ϕ2) ≤ tanh √ tanh cT(ϕ1,ϕ2) . 2 2

The approach exploited above can be extended to other important inequalities, for ex- ample, to exact estimation of the minimal dilatation k0(M) of quasiconformal extensions of M-quasisymmetric homeomorphisms of the circle.

Acknowlegments

I am grateful to the referees for their comments and suggestions which improved the exposition.

References

[1] H. Grunsky: “Koeffizientenbedingungen fur¨ schlicht abbildende meromorphe Funk- tionen”, Math. Z., Vol. 45, (1939), pp. 29–61. [2] R.Kuhnau:¨ “Verzerrungss¨atze und Koeffizientenbedingungen vom Grunskyschen Typ fur¨ quasikonforme Abbildungen”, Math. Nachr., Vol. 48, (1971), pp. 77–105. [3] C. Pommerenke: Univalent Functions, Vandenhoeck & Ruprecht, G¨ottingen, 1975. [4] I.V. Zhuravlev: “Univalent functions and Teichmuller¨ spaces”, Preprint:Inst.of Mathematics, Novosibirsk, 1979, p. 23 (Russian). [5] S.L. Krushkal and R. Kuhnau:¨ Quasikonforme Abbildungen - neue Methoden und Anwendungen, Teubner-Texte zur Math., Bd. 54, Teubner, Leipzig, 1983. [6] R.Kuhnau:¨ “M¨oglichst konforme Spiegelung an einer Jordankurve”, Jber. Deutsch. Math. Verein., Vol. 90, (1988), pp. 90–109. [7] S.L. Krushkal: “Grunsky coefficient inequalities, Carath´eodory metric and extremal quasiconformal mappings”, Comment. Math. Helv., Vol. 64, (1989), pp. 650–660. S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580 579

[8] S.L. Krushkal and R. Kuhnau:¨ “Grunsky inequalities and quasiconformal extension”, Israel J. Math., Vol. 152, (2006), pp. 49–59. [9] R.Kuhnau:¨ “Zu den Grunskyschen Coeffizientenbedingungen”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., Vol. 6, (1981), pp. 125–130. [10] R.Kuhnau:¨ “Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend fur¨ Q-quasikonforme Fortsetzbarkeit?”, Comment. Math. Helv., Vol. 61, (1986), pp. 290– 307. [11] S.L. Krushkal: “Beyond Moser’s conjecture on Grunsky inequalities”, Georgian Math. J., Vol. 12, (2005), pp. 485–492. [12] Y.L. Shen: “Pull-back operators by quasisymmetric functions and invariant metrics on Teichmuller¨ spaces”, Complex Variables, Vol. 42, (2000), pp. 289–307. [13] L. Ahlfors: “An extension of Schwarz’s lemma”, Trans. Amer. Math. Soc., Vol. 43, (1938), pp. 359–364. [14] D. Gaier: Konstruktive Methoden der konformen Abbildung, Springer-Verlag, Berlin- G¨ottingen-Heidelberg, 1964. [15] R.Kuhnau:¨ “Zur Berechnung der Fredholmschen Eigenwerte ebener Kurven”, Z. Angew. Math. Mech., Vol. 66, (1986), pp. 193–200. [16] M. Schiffer: “Fredholm eigenvalues and Grunsky matrices”, Ann. Polon. Math., Vol. 39, (1981), pp. 149–164. [17] G. Schober: “Estimates for Fredholm eigenvalues based on quasiconformal mapping.” In: Numerische, insbesondere approximationstheoretische Behandlung von Funk- tiongleichungen, Lecture Notes in Math., Vol. 333, Springer-Verlag, Berlin, (1973), pp. 211–217. [18] S.E. Warschawski: “On the effective determination of conformal maps”, In: L. Ahlfors, E. Calabi et al. (Eds.): Contribution to the Theory of Riemann surfaces, Princeton Univ. Press, Princeton, (1953), pp. 177–188. [19] L. Ahlfors: “Remarks on the Neumann-Poincar´eintegralequation”,Pacific J. Math. Vol. 2, (1952), pp. 271–280. [20] S.L. Krushkal: “Quasiconformal extensions and reflections”, In: R. Kuhnau¨ (Ed.): Handbook of Complex Analysis: Geometric Function Theory, Vol. II, Elsevier Sci- ence, Amsterdam, 2005, pp. 507–553. [21] K. Strebel: “On the existence of extremal Teichmueller mappings”, J. Anal. Math, Vol. 30, (1976), pp. 464–480. [22] F.P. Gardiner and N. Lakic: Quasiconformal Teichmuller¨ Theory, Amer. Math. Soc., 2000. [23] S. Dineen: The Schwarz Lemma, Clarendon Press, Oxford, 1989. [24] S. Kobayayshi: Hyperbolic Complex Spaces, Springer, New York, 1998. [25] C.J. Earle, I. Kra and S.L. Krushkal: “Holomorphic motions and Teichmuller¨ spaces”, Trans. Amer. Math. Soc., Vol. 944, (1994), pp. 927–948. [26] C.J. Earle and S. Mitra: “Variation of moduli under holomorphic motions”, In: Stony Brook, NY, 1998: The tradition of Ahlfors and Bers, Contemp. Math. Vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39–67. 580 S. Krushkal / Central European Journal of Mathematics 5(3) 2007 551–580

[27] H.L. Royden: “Automorphisms and isometries of Teichmuller¨ space”, Advances in the Theory of Riemann Surfaces (Ann. of Math. Stud.), Vol. 66, Princeton Univ. Press, Princeton, (1971), pp. 369–383. [28] S.L. Krushkal: “Plurisubharmonic features of the Teichmuller¨ metric”, Publications de l’Institut Math´ematique-Beograd, Nouvelle s´erie, Vol. 75, (2004), pp. 119–138. [29] N.A. Lebedev: The Area Principle in the Theory of Univalent Functions,Nauka, Moscow, 1975 (Russian). [30] I.M. Milin: “Univalent Functions and Orthonormal Systems”, Transl. of mathemat- ical monographs, vol. 49, Transl. of Odnolistnye funktcii i normirovannie systemy, Amer. Math. Soc., Providence, RI, 1977. [31] M. Schiffer and D. Spencer: Functionals of finite Riemann Surfaces, Princeton Univ. Press, Princeton, 1954. [32] S. L. Krushkal: “Schwarzian derivative and complex Finsler metrics”, Contemporary Math., Vol. 382, (2005), pp. 243–262. [33] D. Minda: “The strong form of Ahlfors’ lemma”, Rocky Mountain J. Math., Vol. 17, (1987), pp. 457–461. [34] M. Abate and G. Patrizio: “Isometries of the Teichmuller¨ metric”, Ann. Scuola Super. Pisa Cl. Sci., Vol. 26, (1998), pp. 437–452. [35] V. Boˇzin, N. Lakic, V. Markovic and M. Mateljevi´c: “Unique extremality”, J. Anal. Math., Vol. 75, (1998), pp. 299–338. [36] C.J. Earle and Zong Li: “Isometrically embedded polydisks in infinite dimensional Teichmuller¨ spaces”, J. Geom . Anal., Vol. 9, (1999), pp. 51–71. [37] M. Heins: “A class of conformal metrics”, Nagoya Math. J., Vol. 21, (1962), pp. 1–60. [38] S.L. Krushkal: Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979. [39] S.L. Krushkal: “A bound for reflections across Jordan curves”, Georgian Math. J., Vol. 10, (2003), pp. 561–572. [40] R. Kuhnau:¨ “Uber¨ die Grunskyschen Koeffizientenbedingungen”, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, Vol. 54, (2000), pp. 53–60. [41] O. Lehto: Univalent Functions and Teichmuller¨ Spaces, Springer-Verlag, New York, 1987. [42] Yu.G. Reshetnyak: “Two-dimensional manifolds of bounded curvature”, Geometry, IV,EncyclopaediaMath.Sci., Vol. 70, Springer, Berlin, 1993, pp. 3–163, 245–250. Engl. transl. from: Two-dimensional manifolds of bounded curvature, Geometry, 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 189, 273–277, 279 (Russian). [43] H.L. Royden: “The Ahlfors-Schwarz lemma: the case of equality”, J. Anal. Math., Vol. 46, (1986), pp. 261–270.